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This book is a rigorous introduction to real analysis, suitable for a onesemester course at the second-year undergraduate level, based on my experience of teaching this material many times in Australia and Canada. My aim is to give a treatment that is brisk and concise, but also reasonably complete and as rigorous as is practicable, starting from the axioms for a complete ordered field and a little set theory.

Along with epsilons and deltas, I emphasise the alternative language of neighbourhoods, which is geometric and intuitive and provides an introduction to topological ideas. I have included a proper treatment of the trigonometric functions. They are sophisticated objects, not to be taken for granted. This topic is an instructive application of the theory of power series and other earlier parts of the book. Also, it involves the concept of a group, which most students won't have seen in the context of analysis before.

There may be some novelty in the gentle, example-based introduction to metric spaces at the end of the book, emphasising how straightforward the generalisation of many fundamental notions from the real line to metric spaces really is. The goal is to develop just enough metric space theory to be able to prove Picard's theorem, showing how a detour through some abstract territory can contribute back to analysis on the real line.

Needless to say, I claim no originality whatsoever for the material in this book.

The purpose of this course is twofold. First, to give a careful treatment of calculus from first principles. In first-year calculus we learn methods for solving specific problems. We focus on how to use these methods more than why they work. To pave the way for further studies in pure and applied mathematics we need to deepen our understanding of why, as opposed to how, calculus works. This won't be a simple rehashing of first-year calculus at all. Calculus done this way is called real analysis.

In particular, we will consider what it is about the real numbers that makes calculus work. Why can't we make do with the rationals? We will identify the key property of the real numbers, called completeness, that distinguishes them from the rationals and permeates all of mathematical analysis. Completeness will be our main theme through the whole course.

The second goal of the course is to practise reading and writing mathematical proofs. The course is proof-oriented throughout, not to encourage pedantry, but because proof is the only way that mathematical truth can be known with certainty. Mathematical knowledge is accumulated through long chains of reasoning. We can't rely on this knowledge unless we're sure that every link in the chain is sound. In many future endeavours, you will find that being able to construct and communicate solid arguments is a very useful skill.

The real numbers form an ordered field ℝ containing the rationals with an additional property called completeness that the rationals do not satisfy. We need some preliminary definitions to be able to say what completeness means.

2.1. Definition. An upper bound for a subset A ⊂ ℝ is an element b ∈ ℝ such that a ≤ b for all a ∈ A. If A has an upper bound, then A is said to be bounded above.

A lower bound for a subset A ⊂ ℝ is an element b ∈ ℝ such that b ≤ a for all a ∈ A. If A has a lower bound, then A is said to be bounded below.

If A is bounded above and bounded below, then A is said to be bounded.

2.2. Example. Consider the interval [0, 1] = {x ∈ ℝ : 0 ≤ x ≤ 1}. It is bounded above, for example by the upper bound 1. The upper bounds for [0, 1] are precisely the numbers b with b ≥ 1. Thus 1 is the smallest upper bound for [0, 1], and it is of course also the largest element of [0, 1].

Now consider the interval (0, 1) = {x ∈ ℝ :0 < x < 1}, also bounded above, for example by 1. It has the same upper bounds as [0, 1]. Namely, if b ≥ 1 and x ∈ (0, 1), then x < 1 ≤ b, so b is an upper bound for (0, 1).

This is a rigorous introduction to real analysis for undergraduate students, starting from the axioms for a complete ordered field and a little set theory. The book avoids any preconceptions about the real numbers and takes them to be nothing but the elements of a complete ordered field. All of the standard topics are included, as well as a proper treatment of the trigonometric functions, which many authors take for granted. The final chapters of the book provide a gentle, example-based introduction to metric spaces with an application to differential equations on the real line. The author's exposition is concise and to the point, helping students focus on the essentials. Over 200 exercises of varying difficulty are included, many of them adding to the theory in the text. The book is perfect for second-year undergraduates and for more advanced students who need a foundation in real analysis.

Much of the theory developed in Chapters 3, 4, and 5 can be extended to the vastly more general setting of metric spaces. Even if we were only interested in analysis on the real line, this would still be worthwhile. In the following chapter, we will use the abstract theory of this chapter to prove an existence and uniqueness theorem for solutions of differential equations.

9.1. Definition. A metric space is a set X with a function d : X × X → [0, ∞), such that:

We call d a metric or a distance function on X. We sometimes write (X, d) for the set X with the metric d.

It turns out that all we need in order to develop such notions as convergence, completeness, and continuity is the three simple properties that define a metric. Of the three, the triangle inequality is of course the most substantial.

Examples of metric spaces abound throughout mathematics. In the remainder of this section we will explore a few of them. Be sure to verify the three defining properties of a metric if some of the details have been left out.

Let Y be an infinite covering space of a projective manifold M in N of dimension n ≥ 2. Let C be the intersection with M of at most n − 1 generic hypersurfaces of degree d in N. The preimage X of C in Y is a connected submanifold. Let φ be the smoothed distance from a fixed point in Y in a metric pulled up from M. Let φ(X) be the Hilbert space of holomorphic functions f on X such that f2e−φ is integrable on X, and define φ(Y) similarly. Our main result is that (under more general hypotheses than described here) the restriction φ(Y) → φ(X) is an isomorphism for d large enough.

This yields new examples of Riemann surfaces and domains of holomorphy in n with corona. We consider the important special case when Y is the unit ball in n, and show that for d large enough, every bounded holomorphic function on X extends to a unique function in the intersection of all the nontrivial weighted Bergman spaces on . Finally, assuming that the covering group is arithmetic, we establish three dichotomies concerning the extension of bounded holomorphic and harmonic functions from X to .

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